UNIT 21.7, 4.8 Introduction to Functions

HW: 1.7 12 – 24 m3, 264.8 11 – 22

function: A rule that relates two numbers called the input and output. For every input there is exactly one output.

Examples

1.7, 4.8Introduction to FunctionsDefinitions

input: a number entered into a function. (DOMAIN)

output: the result of entering a number into a function. (RANGE)

domain: the collection of all input values for a function (the x-values)

range: the collection of all output values for a function (the y-values and )

1.7, 4.8 Introduction to Functions

Make an input-output table for the function below. Use 1, 2, 3, 4, and 5 as the domain:

Domain Range

Input Output

1 5

2

3

4

5

1.7, 4.8 Introduction to Functions

Make an input-output table for the function below. Use 1, 2, 3, 4, and 5 as the domain:

Domain Range

Input Output

1 0.5

2

3

4

5

1.7, 4.8 Introduction to Functions

Solutiona. Is a function. For every ONE input there is a unique OUTPUT. Domain: Range:Solution“b” Is a NOT a function. The input {1} has two different outputs {5, 7}.

1.7, 4.8 Introduction to Functions

1.7, 4.8 Introduction to FunctionsPractice

Decide whether the graph represents y as a function of x. If yes, then state the domain and range.

Unit 24.3 Quick graph (x, y intercepts)

4.6 Quick graph (slope-intercept)

Homework (due Monday)• 4.3 15 – 54 m3 no 33• 4.6 24 – 45 m3• Use any method to graph. Ignore the

directions.• Two lines per coordinate plane.

4.3 Quick graph (x, y intercepts)4.6 Quick graph (slope-intercept)

• Which method to use depends on how the problem looks when given to you.

• Use x, y intercepts

• Use slope-intercept

Method #2Quick graph using x, y intercepts

• Make an x, y T-table• Make x = 0 and solve for y. (y – intercept)• Make y = 0 and solve for x. (x – intercept)• Plot the two intercepts and draw the line.

Examples

4.6 Graphing lines using the slope and y-intercept

• Slope intercept Form

Slope y – intercept – starting point on the Y- axis

Directions to the next point

y bmx

m

4.6 Graphing y = mx + b

Graph using any method

UNIT 2 4.4 The slope of a line

5.1 writing a linear equation.

𝑚=𝑦2− 𝑦1

𝑥2− 𝑥1

Homework4.4 21 – 41 odd5.1 12 – 22 even write all equations in Slope - intercept form and Standard form

Slope Slope -intercept Standard𝑦=𝑚𝑥+𝑏 𝐴𝑥+𝐵𝑦=𝐶

x

y

run change in x

The slope m of the nonvertical linepassing through the points (x1, y1) and (x2, y2) is

Read y1 as “y sub one”Read x1 as “x sub one”

m = =

(x1, y1)

(x1, y1)

(x2, y2)

(x2, y2)

y2 - y1

(y2 - y1 )(y2 - y1 )

x2 - x1

(x2 - x1 )(x2 - x1 )

FINDING THE SLOPE OF A L INE

(x2, y2)(x1, y1)

rise change in y =

Examples

Examples

Find the slope of the line passing through the given points.

𝑚=𝑦2− 𝑦1

𝑥2− 𝑥1

𝑚=10−41−0

x, y x, y

𝑚=61

𝑚=6

𝑚=𝑦2− 𝑦1

𝑥2− 𝑥1

𝑚=−7−9−6−2

𝑚=−16−8

𝑚=2

4.4 practice

5.1 Writing linear equationsSlope intercept form

Standard formare integers

• Given the slope and the y – intercept , simply substitute slope – intercept form.

• Rewrite the equation into Standard form.

5.1 writing linear equations given slope and y - intercept

• E.g.• Write in slope - intercept form 1)m = 4 and b = 1

slope – intercept formstandard form

You will have TWO answers.

5.1

2) Slope = and the y – intercept = -4

𝑦=𝑚𝑥+𝑏𝑦=−

25𝑥−4 slope – intercept form

25𝑥+𝑦=−4

2 𝑥+5 𝑦=−20 standard form

Unit 25.2, 5.3 writing linear equations

Homework5.2 12 – 33 m35.3 from power point Write all equations in slope – intercept form and standard form and

5.3 homeworkGive your self 5 spaces per problem

1. (3 ,5 ) ,(4 ,10)

7. (4 ,−7 ) ,(3 ,−7)

2 . (4 ,2 ) , (6 ,3) 6. (8 ,−5 ) , (6 ,−4)5. (3 ,5 ) ,(0 ,1)

3 . (−2 ,2 ) ,(0 ,−2)

4. ( 8 ,3 ) ,(3 ,13) 8. (1 ,−3 ) ,(−2 ,−1)

Write a linear equation in slope – intercept form and standard form.

5.2 writing linear equations given slope and a point

Point – slope form m = the slope given

is the point given

Point – slope form

E.g.

Distribute

slope intercept formstandard form

(𝑥 , 𝑦 )

5.2 writing linear equations given slope and a point

1) m = 2, ( 5, 15)

2) m = 3, ( 2, -6)

3) m = , ( -8, 4)

4) m = -2, ( -3, 8)

5) m = , ( 9, -1) 1

2

2

3

5.3 writing linear equations given two points.

• Step 1– Find slope m =

• Step 2 – Substitute m and one of the given points into

point – slope form

2 1

2 1

y y

x x

5.3 writing linear equations given two points.

1) (5, 8), (7, 12)

2) (2, 4), (3, 7)

3) (-1, 8), (2, 2)

Cw 5.2Write an equation of a line.

1) m = 3, (1, 8)

2) m = -1, (2, 4)

3) m = 4, (-1, 3)

4) m = -5, (7, -2)

5) m = , (4, 7)

6) m = -4, (-3, -7)

7) m = 1, (0, -6)

8) m = , (3, -5)

1

2

1

3

3 5y x

6y x

4 7y x

5 33y x

15

2y x

4 19y x

6y x

16

3y x

Hw 5.2, 5.3 Power point Write an equation of a line.

9) (1,6),(4,12)

10) (4,2),(5,5)

11) (-2,4),(1,1)

12) (0,10),(4,2)

13) ( 2,9), (0,13)

14) (8,5),(3,0)

15) (-2,-5), (-4,0)

16) (10,9), (8,9)

2 4

3 10

2

2 10

y x

y x

y x

y x

2 13

3

510

2

9

y x

y x

y

y

Write a linear equation in the form y = mx + b

17) m = 1, b = 5 21) m = -7, (5, -30)

318) m = -2, (3, 6) 22) m = - , b = -4

2

19) (4, 2),(0, 4) 23) (5, 8), (6, 10)

120) (3, 0), (-1, 4) 24) m = , (-

33, 1)